Global existence and blowup for a degenerate and singular parabolic system with nonlocal source and absorptions

Abstract

The aim of this paper is to investigate the asymptotic behavior of positive solutions to the following degenerate and singular parabolic system

$$\begin{array}{lll}{u_t}=(x^ \alpha u_x)_x+ \int \limits_{0}^{a} v^{p_1}{\rm d} x - {k_1} u^{q_1}, \quad 0 < x < a, t > 0, \\ \upsilon_{t}=(x^{\beta} v_{x})_{x} + \int \limits_{0}^{a} u^{p_2}{\rm d}x - k_{2} \upsilon^{q_2}, \, \, \, \, \, 0 < x < a, t > 0,\\ u(0,t) = u(a,t) = \upsilon(0,t) = \upsilon(a,t) = 0, \, \, \, \, \, t > 0,\\ u(x,0) = u_{0}(x) \geq 0, \upsilon(x,0) = \upsilon_{0}(x) \geq 0,\, \, \, \, \, 0 \leq x \leq a,\end{array}$$

where the constants ${0 \leq \alpha, \beta < 1, p_{1}, p_{2}, q_{1}, q_{2}, k_{1}, k_{2} > 0}$. Under appropriate hypotheses, we first prove a local existence of classical solution by a regularization method. Then, we discuss the global existence and blowup of positive solutions by using a comparison principle. Finally, we give the precise blowup rate estimates and the uniform blowup profiles by using the method developed by Souplet [19].